Long range integrable oscillator chains from quantum algebras

TL;DR

This paper constructs and analyzes long-range integrable oscillator chains derived from quantum algebra deformations, revealing their algebraic structure and connections to other integrable systems.

Contribution

It introduces new integrable oscillator chains from quantum algebra deformations and explores their algebraic properties and relations to existing models.

Findings

Explicit deformed Hamiltonians and constants of motion provided

Long-range interactions linked to coalgebra structure

Connection established between oscillator systems and angular momentum chains

Abstract

Completely integrable Hamiltonians defining classical mechanical systems of $N$ coupled oscillators are obtained from Poisson realizations of Heisenberg--Weyl, harmonic oscillator and $sl(2,\R)$ coalgebras. Various completely integrable deformations of such systems are constructed by considering quantum deformations of these algebras. Explicit expressions for all the deformed Hamiltonians and constants of motion are given, and the long-range nature of the interactions is shown to be linked to the underlying coalgebra structure. The relationship between oscillator systems induced from the $sl(2,\R)$ coalgebra and angular momentum chains is presented, and a non-standard integrable deformation of the hyperbolic Gaudin system is obtained.